Adiabatic theorem

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The adiabatic theorem is an important concept in quantum mechanics. Its original form, due to Max Born and Vladimir Fock (1928),^[1] can be stated as follows:

A physical system remains in its instantaneous eigenstate if a given perturbation is acting on it slowly enough and if there is a gap between the eigenvalue and the rest of the Hamiltonian's spectrum.

It may not be immediately clear from this formulation but the adiabatic theorem is, in fact, an extremely intuitive concept. Simply stated, a quantum mechanical system subjected to gradually changing external conditions can adapt its functional form, while in the case of rapidly varying conditions, there is no time for the functional form of the state to adapt, so the probability density remains unchanged.

The consequences of this apparently simple result are many, varied and extremely subtle. In order to make this clear we will begin with a fairly qualitative description, followed by a series of example systems, before undertaking a more rigorous analysis. Finally we will look at techniques used for adiabaticity calculations.

1 Diabatic vs. adiabatic processes
2 Example systems
3 Deriving conditions for diabatic vs adiabatic passage
- 3.1 Diabatic passage
- 3.2 Adiabatic passage
4 Calculating diabatic passage probabilities
- 4.1 The Landau-Zener formula
- 4.2 The numerical approach
5 See also
6 References

[edit] Diabatic vs. adiabatic processes

Diabatic process: Rapidly changing conditions prevent the system from adapting its configuration during the process, hence the probability density remains unchanged. Typically there is no eigenstate of the final Hamiltonian with the same functional form as the initial state. The system ends in a linear combination of states that sum to reproduce the initial probability density.

Adiabatic process: Gradually changing conditions allow the system to adapt its configuration, hence the probability density is modified by the process. If the system starts in an eigenstate of the initial Hamiltonian, it will end in the corresponding eigenstate of the final Hamiltonian.^[2]

At some initial time $\scriptstyle{t_0}$ a quantum mechanical system has an energy given by the Hamiltonian $\scriptstyle{\hat{H}(t_0)}$ ; the system is in an eigenstate of $\scriptstyle{\hat{H}(t_0)}$ labelled $\scriptstyle{\psi(x,t_0)}$ . Changing conditions modify the Hamiltonian in a continuous manner, resulting in a final Hamiltonian $\scriptstyle{\hat{H}(t_1)}$ at some later time $\scriptstyle{t_1}$ . The system will evolve according to the Schrödinger equation, to reach a final state $\scriptstyle{\psi(x,t_1)}$ . The adiabatic theorem states that the modification to the system depends critically on the time $\scriptstyle{\tau = t_1 - t_0}$ during which the modification takes place.

For a truly adiabatic process we require $\scriptstyle{\tau \rightarrow \infty}$ ; in this case the final state $\scriptstyle{\psi(x,t_1)}$ will be an eigenstate of the final Hamiltonian $\scriptstyle{\hat{H}(t_1)}$ , with a modified configuration:

$|\psi(x,t_1)|^2 \neq |\psi(x,t_0)|^2$ .

The degree to which a given change approximates an adiabatic process depends on both the energy separation between $\scriptstyle{\psi(x,t_0)}$ and adjacent states, and the ratio of the interval $\scriptstyle{\tau}$ to the characteristic time-scale of the evolution of $\scriptstyle{\psi(x,t_0)}$ for a time-independent Hamiltonian, $\scriptstyle{\tau_{int} = 2\pi\hbar/E_0}$ , where $\scriptstyle{E_0}$ is the energy of $\scriptstyle{\psi(x,t_0)}$ .

Conversely, in the limit $\scriptstyle{\tau \rightarrow 0}$ we have infinitely rapid, or diabatic passage; the configuration of the state remains unchanged:

$|\psi(x,t_1)|^2 = |\psi(x,t_0)|^2\quad$ .

The so called "gap condition" included in Born and Fock's original definition given above refers to a requirement that the spectrum of $\scriptstyle{\hat{H}}$ is discrete and nondegenerate, such that there is no ambiguity in the ordering of the states (one can easily establish which eigenstate of $\scriptstyle{\hat{H}(t_1)}$ corresponds to $\scriptstyle{\psi(t_0)}$ ). In 1990 J. E. Avron and A. Elgart reformulated the adiabatic theorem, eliminating the gap condition.^[3]

Note that the term adiabatic is traditionally used in thermodynamics to describe processes without the exchange of heat between system and environment (see adiabatic process). The quantum mechanical definition is closer to the thermodynamical concept of a quasistatic process, and has no direct relation with heat exchange. These two different definitions can be the source of much confusion, especially when the two concepts (heat exchange and sufficiently slow processes) are present in a given problem.

[edit] Example systems

[edit] Simple pendulum

As an example, consider a pendulum oscillating in a vertical plane. If the support is moved, the mode of oscillation of the pendulum will change. If the support is moved sufficiently slowly, the motion of the pendulum relative to the support will remain unchanged. A gradual change in external conditions allows the system to adapt, such that it retains its initial character. This is referred to as an adiabatic process.^[4]

[edit] Quantum harmonic oscillator

Main article: Quantum harmonic oscillator

Figure 1. Change in the probability density, $\scriptstyle{|\psi(t)|^2}$ , of a ground state quantum harmonic oscillator, due to an adiabatic increase in spring constant.

The classical nature of a pendulum precludes a full description of the effects of the adiabatic theorem. As a further example consider a quantum harmonic oscillator as the spring constant $\scriptstyle{k}$ is increased. Classically this is equivalent to increasing the stiffness of a spring; quantum mechanically the effect is a narrowing of the potential energy curve in the system Hamiltonian.

If $\scriptstyle{k}$ is increased adiabatically $\scriptstyle{\left(\frac{dk}{dt} \rightarrow 0\right)}$ then the system at time $\scriptstyle{t}$ will be in an instantaneous eigenstate $\scriptstyle{\psi(t)}$ of the current Hamiltonian $\scriptstyle{\hat{H}(t)}$ , corresponding to the initial eigenstate of $\scriptstyle{\hat{H}(0)}$ . For the special case of a system, like the quantum harmonic oscillator, described by a single quantum number, this means the quantum number will remain unchanged. Figure 1 shows how a harmonic oscillator, initially in its ground state, $\scriptstyle{n = 1}$ , remains in the ground state as the potential energy curve is compressed; the functional form of the state adapting to the slowly varying conditions.

For a rapidly increased spring constant, the system undergoes a diabatic process $\scriptstyle{\left(\frac{dk}{dt} \rightarrow \infty\right)}$ in which the system has no time to adapt its functional form to the changing conditions. While the final state must be not identical to the initial state $\scriptstyle{\left(|\psi(t)|^2 = |\psi(0)|^2\right)}$ for a process occurring over a vanishing time period, there is no eigenstate of the new Hamiltonian, $\scriptstyle{\hat{H}(t)}$ , that resembles the initial state. In fact the final state is composed of a linear superposition of many different eigenstates of $\scriptstyle{\hat{H}(t)}$ which sum to reproduce the form of the initial state.

[edit] Avoided curve crossing

Main article: Avoided crossing

Figure 2. An avoided energy level crossing in a two level system subjected to an external magnetic field. Note the energies of the diabatic states, $\scriptstyle{|1\rangle}$ and $\scriptstyle{|2\rangle}$ and the eigenvalues of the Hamiltonian, giving the energies of the eigenstates $\scriptstyle{|\phi_1\rangle}$ and $\scriptstyle{|\phi_2\rangle}$ (the adiabatic states).

For a more widely applicable example, consider a 2-level atom subjected to an external magnetic field.^[5] The states, labelled $\scriptstyle{|1\rangle}$ and $\scriptstyle{|2\rangle}$ using bra-ket notation, can be thought of as atomic angular momentum states, each with a particular geometry. For reasons that will become clear these states will henceforth be referred to as the diabatic states. The system wavefunction can be represented as a linear combination of the diabatic states:

$|\Psi\rangle = c_1(t)|1\rangle + c_2(t)|2\rangle$ .

With the field absent, the energetic separation of the diabatic states is equal to $\scriptstyle{\hbar\omega_0}$ ; the energy of state $\scriptstyle{|1\rangle}$ increases with increasing magnetic field (a low-field seeking state), while the energy of state $\scriptstyle{|2\rangle}$ decreases with increasing magnetic field (a high-field seeking state). Assuming the magnetic field dependence is linear, the Hamiltonian matrix for the system can be written

$\mathbf{H} = \begin{pmatrix} \mu B(t)-\hbar\omega_0/2 & a \\ a^* & \hbar\omega_0/2-\mu B(t) \end{pmatrix}$

where $\scriptstyle{\mu}$ is the magnetic moment of the atom, assumed to be the same for the two diabatic states, and $\scriptstyle{a}$ is some time-independent coupling between the two states. The diagonal elements are the energies of the diabatic states ( $\scriptstyle{E_1(t)}$ and $\scriptstyle{E_2(t)}$ ), however, as $\scriptstyle{\mathbf{H}}$ is not a diagonal matrix, it is clear that these states are not eigenstates of the Hamiltonian.

The eigenvectors of the matrix $\scriptstyle{\mathbf{H}}$ are the eigenstates of the system, which we will label $\scriptstyle{|\phi_1(t)\rangle}$ and $\scriptstyle{|\phi_2(t)\rangle}$ , with corresponding eigenvalues

$\begin{align} \varepsilon_1(t) &= -\frac{1}{2}\sqrt{4a^2 + (\hbar\omega_0 - 2\mu B(t))^2}\\ \varepsilon_2(t) &= +\frac{1}{2}\sqrt{4a^2 + (\hbar\omega_0 - 2\mu B(t))^2}\\ \end{align}$ .

It is important to realise that the eigenstates are the only allowed states of the system; a measurement of the energy of the system at time $\scriptstyle{t}$ will return either $\scriptstyle{\varepsilon_1(t)}$ or $\scriptstyle{\varepsilon_2(t)}$ . The diabatic states $\scriptstyle{|1\rangle}$ and $\scriptstyle{|2\rangle}$ are linear combinations of the eigenstates in which the expectation values of the system energy return the diabatic energies $\scriptstyle{E_1(t)}$ and $\scriptstyle{E_2(t)}$ .

Figure 2 shows the dependence of the diabatic and adiabatic energies on the value of the magnetic field; note that for non-zero coupling the eigenvalues of the Hamiltonian cannot be degenerate, and thus we have an avoided crossing. If an atom is initially in state $\scriptstyle{|\phi_1(t_0)\rangle}$ in zero magnetic field (on the red curve, at the extreme left), an adiabatic increase in magnetic field will ensure the system remains in an eigenstate of the Hamiltonian $\scriptstyle{|\phi_1(t)\rangle}$ (follows the red curve). A diabatic increase in magnetic field will ensure the system follows the diabatic path (the solid black line), such that the system undergoes a transition to state $\scriptstyle{|\phi_1(t_1)\rangle}$ . For finite magnetic field slew rates $\scriptstyle{\left(0<\frac{dB}{dt}<\infty\right)}$ there will be a finite probability of finding the system in either of the two eigenstates. See below for approaches to calculating these probabilities.

These results are extremely important in atomic and molecular physics for control of the energy state distribution in a population of atoms or molecules.

[edit] Deriving conditions for diabatic vs adiabatic passage

We will now pursue a more rigorous analysis.^[6] Making use of bra-ket notation, the state vector of the system at time $\scriptstyle{t}$ can be written

$|\psi(t)\rangle = \sum_n c^A_n(t)e^{-iE_nt/\hbar}|\phi_n\rangle$ ,

where the spatial wavefunction alluded to earlier, is the projection of the state vector onto the eigenstates of the position operator

$\psi(x,t) = \langle x|\psi(t)\rangle$ .

It is instructive to examine the limiting cases, in which $\scriptstyle{\tau}$ is very large (adiabatic, or gradual change) and very small (diabatic, or sudden change).

Consider a system Hamiltonian undergoing continuous change from an initial value $\scriptstyle{\hat{H}_0}$ , at time $\scriptstyle{t_0}$ , to a final value $\scriptstyle{\hat{H}_1}$ , at time $\scriptstyle{t_1}$ , where $\scriptstyle{\tau = t_1 - t_0}$ . The evolution of the system can be described in the Schrödinger picture by the time evolution operator, defined by the integral equation

$\hat{U}(t,t_0) = 1 - \frac{i}{\hbar}\int_{t_0}^t\hat{H}(t^\prime)\hat{U}(t^\prime,t_0)dt^\prime$ ,

which is equivalent to the Schrödinger equation

$i\hbar\frac{\partial}{\partial t}\hat{U}(t,t_0) = \hat{H}(t)\hat{U}(t,t_0)$ ,

along with the initial condition $\scriptstyle{\hat{U}(t_0,t_0) = 1}$ . Given knowledge of the system wave function at $\scriptstyle{t_0}$ , the evolution of the system up to a later time $\scriptstyle{t}$ can be obtained using

$|\psi(t)\rangle = \hat{U}(t,t_0)|\psi(t_0)\rangle$ .

The problem of determining the adiabaticity of a given process is equivalent to establishing the dependence of $\scriptstyle{\hat{U}(t_1,t_0)}$ on $\scriptstyle{\tau}$ .

To determine the validity of the adiabatic approximation for a given process, one can calculate the probability of finding the system in a state other than that in which it started. Using bra-ket notation and using the definition $\scriptstyle{|0\rangle \equiv |\psi(t_0)\rangle}$ , we have:

$\zeta = \langle 0|\hat{U}^\dagger(t_1,t_0)\hat{U}(t_1,t_0)|0\rangle - \langle 0|\hat{U}^\dagger(t_1,t_0)|0\rangle\langle 0|\hat{U}(t_1,t_0)|0\rangle$ .

We can expand $\scriptstyle{\hat{U}(t_1,t_0)}$

$\hat{U}(t_1,t_0) = 1 + {1 \over i\hbar}\int_{t_0}^{t_1}\hat{H}(t)dt + {1 \over (i\hbar)^2}\int_{t_0}^{t_1}dt^\prime\int_{t_0}^{t^\prime}dt^{\prime\prime}\hat{H}(t^\prime)\hat{H}(t^{\prime\prime}) + \ldots$ .

In the perturbative limit we can take just the first two terms and substitute them into our equation for $\scriptstyle{\zeta}$ , recognizing that

${1 \over \tau}\int_{t_0}^{t_1}\hat{H}(t)dt \equiv \bar{H}$

is the system Hamiltonian, averaged over the interval $\scriptstyle{t_0 \rightarrow t_1}$ , we have:

$\zeta = \langle 0|(1 + \frac{i}{\hbar}\tau\bar{H})(1 - {i \over \hbar}\tau\bar{H})|0\rangle - \langle 0|(1 + {i \over \hbar}\tau\bar{H})|0\rangle\langle 0|(1 - {i \over \hbar}\tau\bar{H})|0\rangle$ .

After expanding the products and making the appropriate cancellations, we are left with:

$\zeta = \frac{\tau^2}{\hbar^2}\left(\langle 0|\bar{H}^2|0\rangle - \langle 0|\bar{H}|0\rangle\langle 0|\bar{H}|0\rangle\right)$ ,

giving

$\zeta = \frac{\tau^2\Delta\bar{H}^2}{\hbar^2}$ ,

where $\scriptstyle{\Delta\bar{H}}$ is the root mean square deviation of the system Hamiltonian averaged over the interval of interest.

The sudden approximation is valid when $\scriptstyle{\zeta \ll 1}$ (the probability of finding the system in a state other than that in which is started approaches zero), thus the validity condition is given by

$\tau \ll {\hbar \over \Delta\bar{H}}$ ,

which is, in fact, a statement of the time-energy form of the Heisenberg uncertainty principle.

[edit] Diabatic passage

In the limit $\scriptstyle{\tau \rightarrow 0}$ we have infinitely rapid, or diabatic passage:

$\lim_{\tau \rightarrow 0}\hat{U}(t_1,t_0) = 1$ .

The functional form of the system remains unchanged:

This is sometimes referred to as the sudden approximation. The validity of the approximation for a given process can be determined from the probability that the state of the system remains unchanged:

$P_D = 1 - \zeta\quad$ .

[edit] Adiabatic passage

In the limit $\scriptstyle{\tau \rightarrow \infty}$ we have infinitely slow, or adiabatic passage. The system evolves, adapting its form to the changing conditions,

$|\langle x|\psi(t_1)\rangle|^2 \neq |\langle x|\psi(t_0)\rangle|^2$ .

If the system is initially in an eigenstate of $\scriptstyle{\hat{H}(t_0)}$ , after a period $\scriptstyle{\tau}$ it will have passed into the corresponding eigenstate of $\scriptstyle{\hat{H}(t_1)}$ .

This is referred to as the adiabatic approximation. The validity of the approximation for a given process can be determined from the probability that the final state of the system is different from the initial state:

$P_A = \zeta\quad$ .

[edit] Calculating diabatic passage probabilities

[edit] The Landau-Zener formula

Main article: Landau-Zener transition

In 1932 an analytic solution to the problem of calculating diabatic transition probabilities was published separately by Lev Landau and Clarence Zener^[7], for the special case of a linearly changing perturbation in which the time-varying component does not couple the relevant states (hence the coupling in the diabatic Hamiltonian matrix is independent of time).

The key figure of merit in this approach is the Landau-Zener velocity:

$v_{LZ} = {\frac{\partial}{\partial t}|E_2 - E_1| \over \frac{\partial}{\partial q}|E_2 - E_1|} \approx \frac{dq}{dt}$ ,

where $\scriptstyle{q}$ is the perturbation variable (electric or magnetic field, molecular bond-length, or any other perturbation to the system), and $\scriptstyle{E_1}$ and $\scriptstyle{E_2}$ are the energies of the two diabatic (crossing) states. A large $\scriptstyle{v_{LZ}}$ results in a large diabatic transition probability and vice versa.

Using the Landau-Zener formula the probability, $\scriptstyle{P_D}$ , of a diabatic transition is given by

$\begin{align} P_D &= e^{-2\pi\Gamma}\\ \Gamma &= {a^2/\hbar \over \left|\frac{\partial}{\partial t}(E_2 - E_1)\right|} = {a^2/\hbar \over \left|\frac{dq}{dt}\frac{\partial}{\partial q}(E_2 - E_1)\right|}\\ &= {a^2 \over \hbar|\alpha|}\\ \end{align}$

[edit] The numerical approach

Main article: Numerical solution of ordinary differential equations

For a transition involving a nonlinear change in perturbation variable or time-dependent coupling between the diabatic states, the equations of motion for the system dynamics cannot be solved analytically. The diabatic transition probability can still be obtained using one of the wide variety of numerical solution algorithms for ordinary differential equations.

The equations to be solved can be obtained from the time-dependent Schrödinger equation:

$i\hbar\dot{\underline{c}}^A(t) = \mathbf{H}_A(t)\underline{c}^A(t)$ ,

where $\scriptstyle{\underline{c}^A(t)}$ is a vector containing the adiabatic state amplitudes, $\scriptstyle{\mathbf{H}_A(t)}$ is the time-dependent adiabatic Hamiltonian,^[5] and the overdot represents a time-derivative.

Comparison of the initial conditions used with the values of the state amplitudes following the transition can yield the diabatic transition probability. In particular, for a two-state system:

$P_D = |c^A_2(t_1)|^2\quad$

for a system that began with $\scriptstyle{|c^A_1(t_0)|^2 = 1}$ .

[edit] See also

[edit] References

^ M. Born and V. A. Fock (1928). "Beweis des Adiabatensatzes". Zeitschrift für Physik A Hadrons and Nuclei 51 (3-4): 165-180.
^ T. Kato (1950). "On the Adiabatic Theorem of Quantum Mechanics". Journal of the Physical Society of Japan 5 (6): 435-439.
^ J. E. Avron and A. Elgart (1999). "Adiabatic Theorem without a Gap Condition". Communications in Mathematical Physics 203 (2): 445-463. doi:10.1007/s002200050620.
^ Griffiths, David J. (2005). "10", Introduction to Quantum Mechanics. Pearson Prentice Hall. ISBN 0-13-111892-7.
^ ^a ^b S. Stenholm (1994). "Quantum Dynamics of Simple Systems". The 44th Scottish Universities Summer School in Physics: 267-313.
^ Messiah, Albert (1999). "XVII", Quantum Mechanics. Dover Publications, Inc.. ISBN 0-486-40924-4.
^ C. Zener (1932). "Non-adiabatic Crossing of Energy Levels". Proceedings of the Royal Society of London, Series A 137 (6): 692-702. doi:10.1098/rspa.1932.016510.1098/rspa.1932.0165.

Categories: Quantum mechanics | Physics theorems